Nonradial solutions for semilinear Schrödinger equations with sign-changing potential

Abstract

In this paper, we investigate the existence of infinite nonradial solutions for the Schrödinger equations \begin{equation*} \begin{cases} -\triangle u+b(|x|)u=f(|x|, u), &\quad x\in {\mathbb{R}}^{N},\\ u\in H^{1}({\mathbb{R}}^{N}), \end{cases} \end{equation*} where $b$ is allowed to be sign-changing. Under some assumptions on $b\in C([0,\infty),\mathbb{R})$ and $f\in C([0,\infty)\times\mathbb{R}^{N},\mathbb{R})$, we obtain that the above system possesses infinitely many nonradial solutions. The method of proof relies on critical point theorem.

Keywords

• schrödinger equations
• variational methods
• critical point
• sign-changing potential
• nonradial solution